Analytic expressions and approximations for the on-axis, aberration-free Rayleigh and Debye integral in the case of focusing fields on a circular aperture

We present a derivation of the analytic result for on-axis field values of the Rayleigh diffraction integral, a result that was originally presented in a paper by Osterberg and Smith (1961). The method on which our derivation is based is then applied to other diffraction integrals used in acoustics and optics, e.g., the far-field Rayleigh integral, the Debye integral and the separate near-field part of the Rayleigh integral. Having available our on-axis analytic or semi-analytic solutions for these various cases, we compare the various integrals for wave numbers k pertaining to low-frequency acoustic applications all the way up to high-frequency optical applications. Our analytic results are compared to numerical results presented in the literature.